Combinatorial Inequalities, Matrix Norms, and Generalized Numerical Radii

Abstract

Two new combinatorial inequalities are presented. The main result states that if γ _j, l ≤ j ≤ n, are fixed complex scalars with σ ≡ |Σγ _j| > 0 and δ ≡ max_i,j |γ _i - γ _j | > 0, and if V̰ is a normed vector space over the complex field, then

$${\max _\pi }\left| {{\sum _j}{\gamma _j}{a_{\pi \left( j \right)}}} \right| \geqslant \left[ {\sigma \delta /\left( {2\sigma + \delta } \right)} \right]{\max _j}\left| {{a_j}} \right|,\forall {a_1}, \ldots ,{a_n} \in V$$

π varying over permutations of n letters. Next, we consider an arbitrary generalized matrix norm N and discuss methods to obtain multiplicativity factors for N, i.e., constantsv > 0 such thatvN is submultiplicative. Using our combinatorial inequalities, we obtain multiplicativity factors for certain C-numerical radii which are generalizations of the classical numerical radius of an operator.